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运动方程时程单元先验步长估计初探

袁全 袁驷

袁全, 袁驷. 运动方程时程单元先验步长估计初探[J]. 工程力学, 2022, 39(S): 21-26. doi: 10.6052/j.issn.1000-4750.2021.06.S023
引用本文: 袁全, 袁驷. 运动方程时程单元先验步长估计初探[J]. 工程力学, 2022, 39(S): 21-26. doi: 10.6052/j.issn.1000-4750.2021.06.S023
YUAN Quan, YUAN Si. AN INITIAL STUDY OF A PRIORI ESTIMATION OF STEP-SIZE FOR TIME-ELEMENTS IN SOLVING MOTION EQUATION[J]. Engineering Mechanics, 2022, 39(S): 21-26. doi: 10.6052/j.issn.1000-4750.2021.06.S023
Citation: YUAN Quan, YUAN Si. AN INITIAL STUDY OF A PRIORI ESTIMATION OF STEP-SIZE FOR TIME-ELEMENTS IN SOLVING MOTION EQUATION[J]. Engineering Mechanics, 2022, 39(S): 21-26. doi: 10.6052/j.issn.1000-4750.2021.06.S023

运动方程时程单元先验步长估计初探

doi: 10.6052/j.issn.1000-4750.2021.06.S023
基金项目: 国家自然科学基金项目(51878383,51378293)
详细信息
    作者简介:

    袁驷(1953−),男,北京人,教授,博士,主要从事结构工程研究(E-mail: yuans@tsinghua.edu.cn)

    通讯作者:

    袁全(1993−),男,北京人,博士生,主要从事结构工程研究(E-mail: yuanq19@mails.tsinghua.edu.cn)

  • 中图分类号: O342

AN INITIAL STUDY OF A PRIORI ESTIMATION OF STEP-SIZE FOR TIME-ELEMENTS IN SOLVING MOTION EQUATION

  • 摘要: 基于单元能量投影(element energy projection,EEP)法和边值问题固端法的思想,将其扩展至运动方程问题。该文以单自由度线性元为例,采用Taylor级数渐近展开,对问题的求解进行实质性简化计算;探讨了不经有限元求解便可进行先验定量误差估计的算法;进而实现了自适应单元步长的先验估计和确定。该文给出初步算例,验证了该方法的可行性和有效性。
  • 图  1  例1步长分布图

    Figure  1.  Step-size distribution for Example 1

    图  2  例1单元位移误差比图

    Figure  2.  Displacement errors of elements for Example 1

    图  3  例2步长分布图

    Figure  3.  Step-size distribution for Example 2

    图  4  例2单元误差比图

    Figure  4.  Displacement errors of elements for Example 2

    表  1  有阻尼简谐运动计算数据 (256 s, $ tol = 0.001 $)

    Table  1.   Results of damped harmonic motion (256 s, $ tol = 0.001 $)

    误差估计方法单元数$ {N_{\rm{e}}} $自适应次数$ {N_{{\rm{adp}}}} $先验迭代次数$ {n_{{\rm{iter}}\,0}} $后验迭代次数$ {n_{{\rm{iter}}\,}} $误差比大于1的单元数$ {N_{{\rm{fail}}}} $最小单元长度$ {h_{\,\min }} $最大单元长度$ {h_{\,\max }} $
    直接后验100337347200.092551.21070
    求逆先验1047286315000.090500.75368
    3项先验1044284310400.090500.76208
    2项先验10812943292300.090480.63940
    下载: 导出CSV

    表  2  有阻尼简谐运动计算数据 ( 256s, $ tol = 0.01 $)

    Table  2.   Results of damped harmonic motion (256s, $ tol = 0.01 $)

    误差估计方法单元数$ {N_{\rm{e}}} $自适应次数$ {N_{{\rm{adp}}}} $先验迭代次数$ {n_{{\rm{iter}}\,0}} $后验迭代次数$ {n_{{\rm{iter}}\,}} $误差比大于1的单元数(最大超限值)$ {N_{{\rm{fail}}}} $最小单元长度$ {h_{\,\min }} $最大单元长度$ {h_{\,\max }} $
    直接后验3429916816 (1.384)0.295283.18623
    求逆先验381708741 (1.005)0.280981.30528
    3项先验36485128202 (1.097)0.281031.35135
    2项先验42281116800.279230.99091
    下载: 导出CSV

    表  3  无阻尼自由振动计算数据 ( 256 s, $ tol = 0.001 $)

    Table  3.   Results of undamped free vibration (256 s, $ tol = 0.001 $)

    误差估
    计方法
    单元数$ {N_{\rm{e}}} $自适应次数$ {N_{{\rm{adp}}}} $先验迭代次数$ {n_{{\rm{iter}}\,0}} $后验迭代次数$ {n_{{\rm{iter}}\,}} $误差比大于1的单元数$ {N_{{\rm{fail}}}} $最小单元长度$ {h_{\,\min }} $最大单元长度$ {h_{\,\max }} $
    直接后验279067767700.077490.25854
    求逆先验2790677677000.077490.25845
    3项先验2790677677000.077490.25848
    2项先验2790677677000.077490.26791
    下载: 导出CSV
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    Yuan Si, Yuan Quan, Yan Weiming, Li Yi, Xing Qinyan. New development of solution of equations of motion with adaptive time-step size — Linear FEM based on EEP superconvergence technique [J]. Engineering Mechanics, 2018, 35(2): 13 − 20. (in Chinese) doi: 10.6052/j.issn.1000-4750.2017.05.ST01
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    Yuan Si, Yuan Quan. Fix-end method: A priori quantitative error estimate and control for two-dimensional FEM [J]. Engineering Mechanics, 2021, 38(1): 8 − 14. (in Chinese) doi: 10.6052/j.issn.1000-4750.2020.07.ST05
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出版历程
  • 收稿日期:  2021-06-23
  • 修回日期:  2022-02-15
  • 网络出版日期:  2022-03-11
  • 刊出日期:  2022-06-06

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